Download App

Ratio and Proportion — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsRatio and Proportion3 Marks
Question
If a, b, c are in continued proportion, prove that
$\frac{a^2+b^2+c^2}{(a+b+c)^2}=\frac{a-b+c}{a+b+c}$
✓

Answer

Given, a, b and c are in continued proportion.
Therefore,
$\frac{a}{b}$=$\frac{b}{c}$
$a c=b^2$
LHS $=\frac{a^2+b^2+c^2}{(a+b+c)^2}$
$=\frac{a^2+b^2+c^2+2 a b+2 b c+2 a c-(2 a b+2 b c+2 a c)}{(a+b+c)^2} $
$=\frac{(a+b+c)^2-2\left(a b+b c+b^2\right)}{(a+b+c)^2}$
$ =\frac{(a+b+c)^2-2 b(a+c+b)}{(a+b+c)^2} $
$=\frac{(a+b+c)(a+b+c-2 b)}{(a+b+c)^2} $
$ =\frac{a-b+c}{a+b+c}$
Hence proved.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "[3 marks sum]" from Ratio and Proportion→Ratio and Proportion — all question types→Mathematics STD 10 question papers→ICSE Board STD 10 question papers→ICSE Board question paper generator→

Similar questions

$P$ and $Q$ are the centre of circles of radius $9$ cm and $2$ cm respectively; $P Q=17 \mathrm{~cm} . \mathrm{R}$ is the centre of circle of radius $x \mathrm{~cm}$, which touches the above circles externally, given that $\angle \mathrm{PRQ}=90^{\circ}$. Write an equation in $x$ and solve it.
→
Solve the following equations using the formula
$2 x^2+7 x+5=0$
→
Katrina opened a recurring deposit account with a Nationalised Bank for a period of 2 years. If the bank pays interest at the rate 6% per annum and the monthly instalment is Rs. 1,000, find the:

1) Interest earned in 2 years.

2) Matured value

→
Construct a quadrilateral ABCD in which $\mathrm{AB}=5 \mathrm{~cm}, \mathrm{BC}=4 \mathrm{~cm}, \angle \mathrm{~B}=60^{\circ}, \mathrm{AD}=5.5 \mathrm{~cm}$ and D is equidistant from AB and BC .
→
Find the equation of the line whose inclination is $30^{\circ}$ and which intersects the $y$-axis at the point $(0,-4)$.
→
Which term of the A.P. 3, 10, 17,… will be 84 more than its 13th term?
→
Divide $29$ into two parts so that the sum of the square of the parts is $425.$
→
Mrs. Kapoor opened a Savings Bank Account in State Bank of India on 9th January 2008. Her passbook entries for the year 2008 are given below:
Date Particulars Withdrawals
(in ₹)		 Deposits
(in ₹)		 Balance
(in ₹)
Jan. 9, 2008 By cash - 10,000 10,000
Feb. 12, 2008 By cash - 15,500 25,500
April 6, 2008 To Cheque 3,500 - 22,000
April 30, 2008 To Self 2,000 - 20,000
July 16, 2008 By Cheque - 6,500 26,500
August 4, 2008 To Self 5,500 - 21,000
August 20, 2008 To Cheque 1,200 - 19,800
Dec. 12, 2008 By Cash - 1,700 21,500

Mrs. Kapoor closes the account on 31st December, 2008. If the bank pays interest at 4% per annum, find the interest Mrs. Kapoor receives on closing the account. Give your answer correct to the nearest rupee.
→
Lines $2x – by + 5 = 0$ and $ax + 3y = 2$ are parallel to each other. Find the relation connecting $a$ and $b.$
→
Using the Remainder Theorem, factorise each of the following completely $x^3 + x^2 – 4x – 4$
→